Boundary Stress-Energy Tensor and Newton-Cartan Geometry in Lifshitz Holography

TL;DR

The paper develops a holographic framework for $z=2$ Lifshitz geometries by solving the Lifshitz boundary problem using a vielbein formalism, revealing a novel torsional Newton–Cartan boundary geometry. It constructs the Lifshitz UV completion via frame-fields, derives 4D boundary sources/vevs, and formulates Ward identities in covariant TNC language, including the anisotropic Weyl anomaly with Horava–Lifshitz-like structure. A key result is the emergence of a boundary gauge field intertwined with the TNC structure and the identification of an irrelevant deformation that yields a second UV completion, either Lifshitz UV or hyperscaling-violating AdS-like UV. The work provides a detailed holographic dictionary for Lifshitz holography, clarifying the boundary geometry, conserved currents, and anomaly structure, with potential applications to non-relativistic field theories and Lifshitz hydrodynamics.

Abstract

For a specific action supporting z=2 Lifshitz geometries we identify the Lifshitz UV completion by solving for the most general solution near the Lifshitz boundary. We identify all the sources as leading components of bulk fields which requires a vielbein formalism. This includes two linear combinations of the bulk gauge field and timelike vielbein where one asymptotes to the boundary timelike vielbein and the other to the boundary gauge field. The geometry induced from the bulk onto the boundary is a novel extension of Newton-Cartan geometry that we call torsional Newton-Cartan (TNC) geometry. There is a constraint on the sources but its pairing with a Ward identity allows one to reduce the variation of the on-shell action to unconstrained sources. We compute all the vevs along with their Ward identities and derive conditions for the boundary theory to admit conserved currents obtained by contracting the boundary stress-energy tensor with a TNC analogue of a conformal Killing vector. We also obtain the anisotropic Weyl anomaly that takes the form of a Horava-Lifshitz action defined on a TNC geometry. The Fefferman-Graham expansion contains a free function that does not appear in the variation of the on-shell action. We show that this is related to an irrelevant deformation that selects between two different UV completions.